Quantum Mechanics without Strict Self-Adjointness

Quantum Mechanics without Strict Self-Adjointness

Reducing the number of assumptions in an axiomatic theory is a natural route toward constructing more general frameworks. Historically, the discovery of non-Euclidean geometries provides a well-known example of this approach. Recently, we have applied this strategy to quantum mechanics by abandoning the assumption that the position of a particle must be described by a strictly self-adjoint operator. The remaining principles of quantum theory, together with Galilean symmetry, lead to a generalized quantum framework characterized by a single free parameter with the dimension of length. This mathematical structure preserves Galilean invariance and gives rise to a generalized uncertainty relation that guarantees the existence of a minimal length—an expectation shared by several approaches to quantum gravity. Remarkably, this non-Heisenberg quantum theory yields, without assuming any *a priori* commutation relations, a modified Heisenberg-type uncertainty relation of the form [ \Delta x , \Delta p \geq \sqrt{\frac{\hbar^2}{4} + l_0^2 (\Delta p)^2}, ] which explicitly implies a minimum position uncertainty equal to ( l_0 ). In the limit ( l_0 \to 0 ), the theory smoothly reduces to standard quantum mechanics. By comparing the predictions of this framework with observational data—including the first longitudinal normal modes of the resonant-bar gravitational wave detector **AURIGA** and the (1S!-!2S) transition in hydrogen—we derive upper bounds on the value of the length parameter ( l_0 ).